Two Independent Means: Add Variances, Keep the Populations Clear — Solutions and Rubrics

μ_A and μ_B are the population mean shelf lives in days for packages of A and B; the parameter is A minus B. Use a two-sample t-interval for a difference of population means with unknown σs. Independent SRSs give randomization; 35≤1,000 and 40≤1,200 satisfy each without-replacement bound. Both sample sizes are at least 30, the course large-sample condition. The two samples contain different independently selected packages, not matched observations on the same units.

Point estimate=14.2−12.9=1.3 days. SE=√(2.1²/35+2.4²/40)=√0.27=0.5196152 days. These are separate sample SDs, not known population σs. The supplied technology df≈72.999755 gives t*≈1.992997; df lies between the smaller of 34 and 39 (34) and 35+40−2=73. MOE=t*SE≈1.0355917; interval≈(0.2644083,2.3355917) days. Appropriate t-distribution technology finds df/critical value; a conservative table choice must be stated if used.

We are 95% confident that brand A’s population mean shelf life exceeds B’s by between about 0.264 and 2.336 days. Repeated independent samples of these sizes from the same populations would yield intervals that capture μ_A−μ_B approximately 95% of the time. This one interval may or may not capture the fixed difference. Since zero is excluded and all values are positive, it supports a population mean difference with A higher. The interval describes means, not every package pair.

μ_A and μ_B are population mean battery lives in hours under the same test conditions. Use a two-sample t-test for independent population means. H₀:μ_A−μ_B=0 or μ_A=μ_B. Hₐ:μ_A≠μ_B or μ_A−μ_B≠0. A lower-A question uses μ_A<μ_B or μ_A−μ_B<0; higher-A uses μ_A>μ_B or μ_A−μ_B>0. All hypotheses concern population means, not the observed 8.2 and 7.5.

Independent SRSs establish randomization. 18≤2,000 and 22≤2,500 meet the 10% checks. Both n values are below 30, so inspect both distributions: A is symmetric around 8.2 with balanced counts over 6.2–10.2; B is symmetric around 7.5 with balanced counts over 5.5–9.5. Neither has an isolated extreme. When either sample is small, both sample distributions should be free of strong skewness and outliers; looking only at A would leave B’s condition unsupported.

SE=√(1.13759292²/18+1.08012345²/22)≈0.35344834 hours. t=[(8.2−7.5)−0]/SE≈1.9804875. Under equal population means it has an approximate t distribution with appropriate Welch df≈35.633502, found by technology. df is between min(17,21)=17 and 18+22−2=38. The not-equal alternative uses both theoretical tails, giving p≈2(0.02770495)=0.05540989. A suitable t table can give a bound or an explicitly conservative df approximation; do not use standard normal or one tail silently.

Assuming equal population mean battery life under these conditions, the probability of a standardized sample mean difference at least 1.9805 in absolute value is approximately 0.05541. Since this is >0.05, fail to reject H₀. There is not convincing statistical evidence that the model population mean battery lives differ. This evidence statement concerns population averages, not a guarantee comparing every A/B battery, and it does not prove either equality or a universal difference.

A: independent SRSs, fractions below 10%, both n≥30 support the course large-sample route. B: random/fractions satisfied, both populations approximately Normal and both observed samples clean support small samples. C: random/fractions satisfied; because either n is below 30, both sample distributions must be checked and both are clean. D fails that same branch: a clean small group does not compensate for strong skew/outlier in the other group. E has randomization and adequate n, but 120>100 violates the first population’s sampling fraction; 120≤300 for the second is not enough. F’s random assignment supplies experiment randomization, population-sampling fraction is unnecessary for assignment, and both n40 meet the sample-data condition. Volunteer participation limits broad population generalization even though causal treatment comparison is possible under the design.

Use a two-sample t-interval and a two-sample t-test for independent mean reaction times. Let μ_music and μ_reading be mean reaction time in milliseconds under the respective treatments for a scope justified by these adult volunteers; estimate or test μ_music−μ_reading. Random assignment can support a causal treatment effect within this design, while volunteering does not make participants representative of all people. If each volunteer receives both treatments, responses within that person are dependent; compute music-minus-reading differences and use a one-sample paired t method rather than adding two independent sample variances.

Study 1: we are 95% confident A’s population mean wait is 1.2 minutes lower to 0.8 minutes higher than B’s; zero is included, so a difference lacks convincing interval support. Study 2: we are 95% confident A’s mean wait is 0.4 to 2.0 minutes higher; zero is excluded, supporting a population mean difference with A higher. Repeated independent samples of the same sizes/populations would give intervals capturing the fixed mean difference about 95% of the time; an individual interval may or may not capture it. Reversing B−A gives (−0.8,1.2) and (−2.0,−0.4), preserving information and zero inclusion.

Assuming equal store population mean waits, the probabilities of standardized differences at least as extreme in either direction are 0.48 and 0.008. Study 1: p>0.05, fail to reject; insufficient evidence of a population mean difference. Study 2: p≤0.05, reject; convincing evidence of a mean difference. The failure decision does not prove equality; the study may simply not distinguish the values convincingly. Both conclusions concern store population mean waits, not every customer’s wait.